\( \newcommand{\nsp}[1]{{\mathcal N}\left(#1\right)} \newcommand{\col}[1]{{\mathcal C}\left(#1\right)} \newcommand{\row}[1]{{\mathcal R}\left(#1\right)} \newcommand{\transpose}[1]{#1^t} \newcommand{\vect}[1]{{\mathbf #1}} \newcommand{\twopartcol}[2]{\left[\begin{array}{c}#1\\\hline#2\end{array}\right]} \newcommand{\twopartrow}[2]{\left[\begin{array}{c|c}#1&#2\end{array}\right]} \)

Extended Echelon Form and Four Subspaces

Robert A. Beezer
Department of Mathematics and Computer Science
University of Puget Sound

August 21, 2013
Revised: October 23, 2013; November 5, 2013 1 

Associated with any matrix, there are four fundamental subspaces: the column space, row space, (right) null space and left null space. We describe a single computation that makes readily apparent bases for all four of these subspaces. Proofs of these results rely only on matrix algebra, not properties of dimension. A corollary is the equality of column rank and row rank.


Jeff Stuart provided a literature search of similar results in introductory textbooks, of which there was only one. Helpful suggestions from a referee have improved this note, and are greatly appreciated.